|Imam Jauhari Maknun||Department of Civil and Environmental Engineering, Faculty of Engineering, Universitas Indonesia, Kampus UI Depok, Depok 16424, Indonesia|
|Salfa Zarfatina||Department of Civil and Environmental Engineering, Faculty of Engineering, Universitas Indonesia, Kampus UI Depok, Depok 16424, Indonesia|
and computational methods are essential to support the development of
infrastructure. Composite material has been used in many applications; in the
laminated composite, failure due to excessive interlaminar stresses between two
materials causes delamination. Thus, functionally graded materials (FGMs) have
emerged. A numerical computation such as the finite element method (FEM) is
widely used to support the analysis of FGMs in structural applications. The
discrete shear gap DSG element is developed using Timoshenko beam theory, where
the shear correction factor is used in their formulation. The shear correction
factor is assumed to be constant in many applications; thus, it is valid for
isotropic homogenous material. However, the effect of shear deformation
significantly impacts the results of the FGMs beam, so the shear correction
factor cannot be considered constant. Therefore, this paper presents the shear correction factor effect on static analysis of
FGMs beam using DSG element. Various boundary conditions with length thickness
ratio (L/h = 4) are evaluated. The DSG element yields good results in
FGMs beam for different power-law index ratios. Furthermore, the DSG element
result shows that the higher the modulus of elasticity ratio of the top-to-bottom
material, the further the difference between k FGMs and k = 5/6
(constant). The DSG element can provide precise results without shear locking.
Composite; DSG; FEM; FGMs; Shear correction factors
Composites are the combination of materials selected based on the variety of the physical properties of each constituent material. They are combined to produce a new material with unique properties compared to those of the primary material. The application of composite material in beam structures can be found in Benbouras et al. (2017). Delamination remains a large problem in laminated composites (Reddy, 2006). In an attempt to solve this problem, new composite materials, called Functionally Graded Materials (FGMs), were first proposed in 1984 by a group of materials scientists in Japan.
The material properties vary gradually and continuously from one surface to another according to the given function. The gradation of material properties through thickness avoids sudden changes in stress distribution. In general, this material is a mixture of ceramics on the top and metal at the bottom. The main benefit of using this material is that it can withstand extreme situations like a high-temperature environment. The use of this material increases rapidly in many applications. Numerical analysis such as FEM is desired to support the development of FGMs.
FEM is one of the numerical techniques used to support structural analysis. Many papers deal with the recent development of FEM in plate and shell structures for isotropic material (Katili et al., 2014; Maknun et al., 2016; Irpanni et al., 2017; Katili et al., 2017; Katili et al., 2018a; Katili et al., 2018b; Katili et al., 2019a). The results of FEM for the composite material can be found in (Katili et al., 2015; Maknun et al., 2015; Katili et al., 2018c; Katili et al., 2018d; Katili et al., 2019b; Maknun et al., 2020). From the results, FEM can support the analysis of plate and shell structures in isotropic and composite materials with accurate results.
FEM performs very well in calculating FGMs beam (Simsek, 2009; Thai and Vo 2012; Nguyen et al., 2013, Aghazadeh et al., 2014; Jing et al., 2016; Katili and Katili 2020). Many beam elements have been proposed for application in beam structures using either Euler Bernoulli's or Timoshenko’s (Timoshenko, 1921; Timoshenko, 1922) beam theories. The beam element developed using Euler Bernoulli's beam theory needs C1 continuity and can only be used for thin beam structures. Timoshenko beam elements were proposed to overcome these limitations. Beam elements developed from Timoshenko beam theory can be used for thin to thick beam structures and need only C0 continuity. However, the shear locking phenomenon exists in Timoshenko beam elements. There are several methods to overcome this phenomenon. One of these is the discrete shear gap (DSG) method proposed by Bletzinger et al. (2000).
For the beam element developed from Timoshenko beam theory, such as the DSG element, the shear correction factor is one of the necessary terms. This factor obtained from the shear strain energy from equilibrium equations equated to the shear strain energy obtained from the constitutive equation (Meena et al., 2012). In isotropic rectangular beams, this factor is equal to k = 5/6. For many applications of FGMs in beam structures, the shear correction factor is assumed to be constant. This factor is not constant for non-isotropic material like FGMs. This factor has a significant effect when analyzing thick beam problems in which the shear energy is crucial, which many papers have dealt with. (Timoshenko, 1922; Nguyen et al., 2008; Hosseini-Hashem et al. 2010).
This paper aims to study the effect of the shear correction factor on the FGMs beam by using the DSG element. The equation for the shear correction factor is taken from Meena et al. (2012). FGMs beams with various boundary conditions and length thickness ratios (L/h = 4) are evaluated. Different power-law indexes (n) are also assessed to understand the convergence behavior of the DSG element in the FGMs beam.
the analysis, it can be concluded that the performance of DSG elements on FGMs
beams yields good results. The
convergence results for k = 5/6 and k FGMs are close to the HOSDT reference
for the number of elements ? 16. The top and bottom elastic modulus (Et/Eb)
ratio and the power-law index (n) influence the shear correction factor.
The greater the Et/Eb ratio, the greater the effect of the shear correction factor. In the
case with Et/Eb = 0.35 for thick beams (L/h
= 4), the error of displacement ranged from 0.184% (clamped-free) to 1.394%
(clamped-clamped). In the case with Et/Eb = 20 for thick beams (L/h = 4),
the error of displacement ranged from 4.432% (clamped-free) to 25.751%.
(clamped-clamped). The largest error occurred in the clamped-clamped support
shear energy is more important here than in other supports.
financial support from Penelitian Dasar Unggulan Perguruan Tinggi 2021 Nomor:
NKB-199/UN2.RST/HKP.05.00/2021 is gratefully acknowledged.
Aghazadeh, R., Cigeroglu, E., Dag, S., 2014. Static and Free Vibration Analyses of Small-Scale Functionally Graded Beams Possessing a Variable Length Scale Parameter using Different Beam Theories. European Journal of Mechanics - A/Solids, Volume 46, pp. 1–11
Benbouras, Y., Maziri, A., Mallil, E., Echaabi, J., 2017. A Nonlinear Analytical Model for Symmetric Laminated Beams in Three-point Bending. International Journal of Technology, Volume 8(3), pp. 437–447
Bletzinger, K.U., Bischoff, M., Ramm, E., 2000. A Unified Approach for Shear-Locking-Free Triangular and Rectangular Shell Finite Elements. Computers & Structures, Volume 75 (3), pp. 321–334
Hosseini-Hashemi, Sh., Taher, R.D., Akhavan, H., Omidi, M., 2010. Free Vibration of Functionally Graded Rectangular Plates using First Order Shear Deformation Plate Theory. Applied Mathematical Modelling, Volume 34(5), pp. 1276–1291
Irpanni, H., Katili, I., Maknun, I.J., 2017. Development DKMQ Shell Element with Five Degrees of Freedom per Nodal. International Journal of Mechanical Engineering and Robotics Research, Volume 6(3), pp. 248–252
Jing, L.L., Ming, P.J., Zhang, W.P., Fu, L.R., Cao, Y.P., 2016. Static and Free Vibration Analysis of Functionally Graded Beams by Combination Timoshenko Theory and Finite Volume Method. Composite Structures, Volume 138, pp. 192–213
Katili, I., Batoz, J.L., Maknun, I.J., Hamdouni, A., Millet, O., 2014. The Development of DKMQ Plate Bending Element for Thick to Thin Shell Analysis based on Naghdi/Reissner/Mindlin Shell Theory. Finite Elements in Analysis and Design, Volume 100, pp. 12–27
Katili, I., Maknun, I.J., Tjahjon,o E., Alisjahbana, I., 2017. Error Estimation for the DKMQ24 Shell Element using Various Recovery Methods. International Journal of Technology, Volume 8(6), pp. 1060–1069
Katili, I., Batoz, J.L., Maknun, I.J., and Lardeur, P., 2018a. A Comparative Formulation of DKMQ, DSQ and MITC4 Quadrilateral Plate Elements with New Numerical Results based on S-Norm Tests. Computer & Structure, Volume 204, pp. 48–64
Katili, A.M., Maknun, I.J., Katili, I., 2018b. Theoretical Equivalence and Numerical Performance of T3s and MITC3 Plate Finite Elements. Structural Engineering and Mechanics, Volume 69(5), pp. 527–536
Katili, I., Maknun, I.J., Batoz, J.L., Katili, A.M., 2019a. A Comparative Formulation of T3?s, DST, DKMT and MITC3+ Triangular Plate Elements with New Numerical Results based on S-Norm Tests. European Journal of Mechanics, A/Solids, Volume 78, https://doi.org/10.1016/j.euromechsol.2019.103826
Katili, I., Maknun, I.J., Millet, O., Hamdouni, A., 2015. Application of DKMQ Element for Composite Plate Bending Structures. Composite Structures, Volume 132, pp. 166–174
Katili, I., Maknun, I.J., Batoz, J.L., Ibrahimbegovi?, A., 2018c. Shear Deformable Shell Element DKMQ24 for Composite Structures. Composite Structures, Volume 202, pp. 182–200
Katili, I., 2017. Unified and Integrated Approach in a New Timoshenko Beam Element. European Journal of Computational Mechanics, Volume 26(3), pp. 282–308
Katili, I., Maknun, I.J., Batoz, J.L., Katili, A.M., 2018d. Asymptotic Equivalence of DKMT and MITC3 Elements for Thick Composite Plate. Composite Structures, Volume 206, pp. 363–379
Katili, I., Maknun, I.J., Katili, A.M., Bordas, S.P.A., Natarajan, S., 2019b. A Unified Polygonal Locking-Free Thin/Thick Smoothed Plate Element. Composite Structures, Volume 219, pp 147–157
Katili, A.M., Katili, I., 2020. A Simplified UI Element using Third-Order Hermitian Displacement Field for Static and Free Vibration Analysis of FGM Beam. Composite Structures, Volume 250, https://doi.org/10.1016/j.compstruct.2020.112565
Maknun, I.J., Katili, I., Millet, O., Hamdouni, A., 2016. Application of DKMQ24 Shell Element for Twist ff Thin-Walled Beams: Comparison with Vlasov Theory. International Journal for Computational Methods in Engineering Science and Mechanics, Volume 17(6), pp. 391–400
Maknun, I.J., Katili, I., Ibrahimbegovi?, A., Katili, A.M., 2020. A New Triangular Shell Element for Composites Accounting for Shear Deformation. Composite Structures, Volume 243, https://doi.org/10.1016/j.compstruct.2020.112214
Maknun, I.J., Katili, I., Purnomo, H., 2015. Development of DKMT Element for Error Estimation in Composite Plate Structures. International Journal of Technology, Volume 6(5), pp. 780–789
Malikan, M., Eremeyev, V.A., 2020. A New Hyperbolic-Polynomial Higher-Order Elasticity Theory for Mechanics of thick FGM Beams with Imperfection in the Material Composition. Composite Structures, Volume 249, 112486
Meena R., Tounsi, A., Mouaici, F., Mechab, I., Zidi, M., Bedia, E.A., 2012. Analytical Solutions for Static Shear Correction Factor of Functionally Graded Rectangular Beams. Mechanics of Advanced Materials and Structures, Volume 19(8), pp. 641–652
Nguyen, T-K., Vo, Thuc, P., Thai, H-T., 2013. Static and Free Vibration of Axially Loaded Functionally Graded Beams based on the First-Order Shear Deformation Theory. Composites Part B: Engineering, Volume 55, pp. 147–57
Nguyen, T-K., Sab, K., Bonnet, G., 2008. First-Order Shear Deformation Plate Models for Functionally Graded Materials. Composite Structures, Volume 83(1), pp. 25–36
Reddy, J.N., 2006. An Introduction to the Finite Element Method. New York: McGraw-Hill
Simsek, M., 2009. Static Analysis of a Functionally Graded Beam Under a Uniformly Distributed Load by Ritz Method. International Journal of Engineering and Applied Sciences (IJEAS), Volume 1(3), pp. 3–4
Timoshenko, S., 1922. On the Transverse Vibrations of Bars of Uniform Crosssection. Philosophical Magazine, Volume 43(253), pp. 125–131
Timoshenko, S., 1921. On the Correction for Shear of Differential Equation for Transverse Vibrations of Prismatic Bars. Philosophical Magazine, Volume 41(245), pp. 744–746
H-T., Vo, T.P., 2012. Bending and Free Vibration of Functionally Graded Beams using
Various Higher-Order Shear Deformation Beam Theories. International Journal of Mechanical Sciences, Volume 62(1), pp.